Shortest path cost distribution in random graphs with positive integer edge costs

The probability distribution of the shortest path cost from a source node to an arbitrary destination node is considered for a random network model consisting of a complete digraph with positive integer random edge costs. Edge costs are chosen according to a common probability distribution for each direction. For this model, the joint distribution of the number of nodes which have a given sequence of shortest path costs from an arbitrary source node is determined explicitly. An expression is then obtained for the distribution of the shortest path cost between two arbitrary nodes using this joint distribution. The main result is the derivation of tight bounds and a sharp limit result for the distribution of the shortest path cost as the number of nodes tends to infinity. Numerical examples are presented to illustrate these results